Final answer:
To find the probability that a random sample of protozoa will have a mean life expectancy of 46 or more days, we can use the Central Limit Theorem. By calculating the z-score and using a standard normal distribution table or calculator, we find that the probability is approximately 6.71%.
Step-by-step explanation:
The question is asking us to find the probability that a simple random sample of 25 protozoa will have a mean life expectancy of 46 or more days. We are given that the distribution of the life expectancies of the protozoa is normal with a mean of 43 days and a standard deviation of 10.1 days.
To solve this problem, we can use the Central Limit Theorem. According to the Central Limit Theorem, the distribution of sample means will be approximately normal, regardless of the shape of the population distribution, as long as the sample size is sufficiently large (n >= 30).
We can calculate the z-score for a sample mean of 46 days using the formula: z = (x - μ) / (σ / sqrt(n)), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. Substituting the given values into the formula, we have z = (46 - 43) / (10.1 / sqrt(25)) = 3 / 2.02 = 1.49.
Using a standard normal distribution table or a calculator, we can find the probability that a z-score is greater than or equal to 1.49. This probability represents the probability of obtaining a sample mean of 46 or more days. The probability is approximately 0.0671 or 6.71%.